Fluctuational internal Josephson effect in topological insulator film
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J u l Fluctuational internal Josephson effect in topological insulator film
D.K. Efimkin and Yu.E. Lozovik
1, 2, ∗ Institute of Spectroscopy RAS, 142190, Troitsk, Moscow, Russia Moscow Institute of Physics and Technology, 141700, Moscow, Russia
Tunneling between opposite surfaces of topological insulator thin film populated by electrons andholes is considered. We predict considerable enhancement of tunneling conductivity by Cooperelectron-hole pair fluctuations that are precursor of their Cooper pairing. Cooper pair fluctuationslead to the critical behavior of tunneling conductivity in vicinity of critical temperature with criticalindex ν = 2. If the pairing is suppressed by disorder the behavior of tunneling conductivity in vicinityof quantum phase transition is also critical with the index µ = 2. The effect can be interpreted asfluctuational internal Josephson effect and it is general phenomenon for electron-hole bilayers. Thepeculiarities of the effect in other realizations of electron-hole bilayer are discussed. PACS numbers: 71.35.Lk, 74.50.+r, 74.40.Gh
I. INTRODUCTION
Cooper pairing of spatially separated electrons andholes was predicted in the system of semiconductor quan-tum wells more then thirty years ago . Later it wasobserved in quantum Hall bilayer at total filling factor ν T = 1 that can be presented as the system of spatiallyseparated composite electrons and composite holes (see and references therein). After graphene discovery Cooperpairing of Dirac electrons and holes in the structure of in-dependently gated graphene layers has been proposed .Recently possibility of Cooper pairing of Dirac electronsand holes was predicted in thin film of topological in-sulator (TI), new unique class of solids that has topologi-cally protected Dirac surface states . The electron-holepairing in that system is the realization of topologicalsuperfluidity and hosts Majorana fermions on edges andvortices that is the topic of extraordinary interest dueto possibility to use them in quantum computation .Also the Cooper pairing can lead to a number of interest-ing physical effects including superfluidity , anomalousdrag effect , nonlocal Andreev reflection .The most prominent manifestation of electron-holepairing is internal Josephson effect . The coherencebetween electron and hole states leads to a tunnel cur-rent j T = j sin( φ − φ T ) that depends on the phase φ of electron-hole condensate. Here j is maximal value ofthe current carried by the condensate and φ T is the phaseof the tunnel matrix element. Dynamic of the phase onthe macroscopic scale is described by the action with La-grangian analogous to the one for the superconductingJosephson junctions. But current-voltage characteristicsof the electron-hole bilayer drastically differs from onesof the latter. In equilibrium state the phase of the or-der parameter is fixated φ = φ T and the tunnel currentis zero hence in electron-hole system there is no ana-log of DC Josephson effect. The coherent tunnel currentflows in the non equilibrium state driven by a voltagebias between the electron and hole layers. It leads to acolossal enhancement of tunneling conductivity at zerovoltage bias. The effect has been observed in quantum Hall bilayer and its microscopical and macroscopi-cal description were addressed in a number of interestingtheoretical papers .Cooper electron-hole pairing can appear above criti-cal temperature as thermodynamic fluctuations. Partic-ulary they lead to the logarithmic divergence of a dragconductivity as a function of a temperature and apseudogap formation in single-particle density of states ofelectrons and holes . Manifestations of Cooper pair fluc-tuations in tunneling have not been consider previously.Since tunneling conductivity is colossally enhanced in thepaired state one can anticipate its strong enhancementby Cooper pair fluctuations above critical temperaturein analogy with strong contribution of electron-electronCopper pair fluctuations in superconductor to its dia-magnetic susceptibility and electric conductivity . In-deed, tunneling current in electron-hole bilayer can betransferred by Cooper pair fluctuations. Since the am-plitude of pairing fluctuations increases in a vicinity ofcritical temperature, as fluctuations of an ordered statedo for different phase transitions, one can expect the sig-nificant enhancement of tunneling conductivity and itscritical behavior. The described effect can be called fluc-tuational internal Josephson effect and it is rather generalphenomenon for electron-hole bilayers. Here we developthe microscopic theory of the effect and its macroscopictheory will be published elsewhere. We have consideredthe effect in topological insulator thin film and its pecu-liarities in other realizations of electron-hole bilayer arediscussed in Conclusions.The rest of the paper is organized as follows. In Section2 we briefly discuss the model used for the description ofinteracting electrons and holes in TI film. In Section 3the microscopical description of Cooper pair fluctuationsis introduced. In Section 4 the tunneling conductivity be-tween the opposite surfaces of topological insulator filmis calculated and Section 5 is devoted to the analysis ofresults and conclusions. FIG. 1. (Color online) a) Experimental setup for measure-ments of tunneling conductivity between opposite surfaces ofTI populated by electrons and holes. b) Dispersion laws ofthe electrons and holes in equilibrium. Dashed line denoteselectrochemical potential of TI surfaces. c) Dispersion law ofthe electrons and holes in nonequilibrium state induced byvoltage bias V.
II. THE MODEL
The setup for the experimental investigation of tun-neling conductivity between spatially separated electronsand holes in TI film is presented on Fig.1. Voltage V eh between the external gates induces equilibrium concen-trations of electrons and holes on the opposite surfaces.Voltage V drives the system from equilibrium and in-duces charge current between the layers. If the side sur-faces of the film are gapped, for example, by orderedmagnetic impurities introduced to TI surface the chargecan be transferred only via interlayer tunneling and thetunneling resistance can be measured. Also charge trans-port through TI side surfaces is unimportant if the areaof the tunneling junction is large enough.Possibility and peculiarities of electron-hole Cooperpairing in the TI film in realistic model that takes intoaccount screening, disorder and interlayer tunneling hasbeen considered in our paper . Here we focus on investi-gation of Cooper pair fluctuations and their role in tun-neling. Hamiltonian of the system H = H eh + H d + H T in-cludes kinetic and electron-hole interaction energies H eh ,interaction with disorder H d and the part describing tun-neling H T . The first part in single-band approximationthat ignores valence (conduction) band on the surfacewith excess of electrons (holes) is given by H eh = X p ξ p a + p a p − X p ξ p b + p b p ++ X pp ′ q U ( q )Λ p + q , pp ′ − q , p ′ a + p + q b + p ′− q b p ′ a p . (1)Here a p is annihilation operator for a electron on thesurface with excess of electrons and b p is annihilationoperator for a electron on the surface with excess ofholes ; ξ p = v F p − E F is Dirac dispersion law inwhich v F and E F are velocity and Fermi energy ofelectrons and holes. We consider the balanced case a) Γ c = + Γ c b) T T + + T T + + + T T + + T T + FIG. 2. a) Diagrammatic representation of the Bethe-Salpeterequation for the Cooper propogator Γ c ; b) Feynman diagramsfor a tunnel conductivity. Solid (dashed) line corresponds toelectrons on the surface of TI film with excess of electrons(holes). since it is favorable for Cooper pairing and the pair-ing is sensitive to concentration mismatch of electronsand holes. U ( q ) is screened Coulomb interaction be-tween electrons and holes (see for its explicit value) andΛ p + q , pp ′ − q , p ′ = cos ( φ p , p + q /
2) cos ( φ p ′ , p ′ + q /
2) is angle factorthat comes from the overlap of spinor wave functions oftwo-dimensional Dirac fermions. Critical temperature ofpairing in Bardeen-Cooper-Schrieffer (BCS) theory thatignores disorder and tunneling is given by T = 2 γ ′ E F π exp − /ν F U ′ (2)where U ′ is Coulomb coupling constant ; γ ′ = e C where C ≈ ,
577 is the Euler constant.We do not specify explicitly the interaction Hamilto-nian with disorder H d since both short-range and long-range Coulomb impurities lead to pairbreaking and cansuppress Cooper pairing. Short-range disorder scattersonly one component of Cooper pair since they are spa-tially separated and long-range Coulomb impurities actsdifferently on components of Cooper pair since they havedifferent charge. Below we introduce phenomenologicaldecays of electrons γ a and holes γ b .The tunneling of electrons between the opposite sur-faces of topological insulator thin film with conservingmomentum can be described by the following Hamilto-nian H T = T + T + = X p (cid:0) tb + p a p + t ∗ a + p b p (cid:1) , (3)where t is the tunneling amplitude. We consider influenceof tunneling on pairing to be weak and treat it below asperturbation.The described model is applicable for description oftunneling in TI films which width is larger than value atwhich t ≈ T . If t ≫ T tunneling strongly influenceselectron-hole pairing and it can not be treated as pertur-bation. Particularly it induces electron-hole condensatewith fixated phase and smears critical temperature to thepaired state. Our calculation shows that the describedmodel is applicable for thin films of Bi Se at d >
10 nm.In that case critical temperature without disorder canachieve T ≈ . E F = 5 meV and the pairing isnot suppress by disorder if electrons and holes have ex-ceptional hight mobilities of order µ ∼ sm / Vs. Itshould be noted that single-band and static screening ap-proximations used for calculation of critical temperature usually underestimate critical temperature of Cooperpairing between Dirac particles . III. COOPER PAIR FLUCTUATIONS
For the microscopical description of Cooper pair fluctu-ations we introduce Cooper propagator Γ Rc ( ω ). It corre-sponds to the two-particle vertex function in the Cooperchannel and satisfies the Bethe-Salpeter equation de-picted on Fig. 2 (a). In Bardeen-Cooper-Schrieffer (BCS)approximation its solution can be presented in the formΓ Rc ( ω ) = U ′ − U ′ Π Rc ( ω ) , (4)where Π Rc ( ω ) corresponds to electron-hole bubble dia-gram that can be interpreted as Cooper susceptibility ofthe system. After direct calculation it can be presentedin the following formΠ Rc ( ω ) = 1 U ′ − ν F ln TT − ν F Ψ (cid:18) (cid:19) −− ν F Ψ (cid:18) − iω πT + γ πT (cid:19) . (5)Here γ = ( γ a + γ b ) / ; ν F is thedensity of states of electrons and holes on the Fermi level;Ψ( x ) is the digamma function. Cooper pair propagatoracquires the following formΓ Rc ( ω ) = 1 ν F TT + Ψ (cid:0) − i ω πT + γ πT (cid:1) − Ψ (cid:0) (cid:1) . (6)In the absence of disorder Γ Rc ( ω ) = 0 at the critical tem-perature T indicating Cooper instability of the systemagainst Cooper pairing. Critical temperature for disor-dered system T d at which Γ Rc ( ω ) = 0 satisfies the follow-ing equationln T d T + Ψ (cid:18)
12 + γ πT (cid:19) − Ψ (cid:18) (cid:19) = 0 . (7)This equation has nontrivial solution if γ < γ , where γ = 0 . T is the critical Cooper pair decay value. In -4 -2 0 2 410 -3 -2 -1 c T V (mV)mm- ) BCS
FIG. 3. (Color online) Tunneling conductivity σ as a func-tion of bias voltage V for noninteracting (dashed line) andinteracting (solid line) electrons and holes for γ = 0 . T = 0 . opposite case the pairing is suppressed by disorder. Thevalue γ corresponds to quantum critical point at zerotemperature.Above critical temperature T d the expression forCooper pair propagator (6) at ω → Rc ( ω ) = 1 ν F πT d πT d ln TT d − iω Ψ ′ ( + γ πT d ) . (8)If the pairing is suppressed by the disorder Cooper pairpropagator at zero temperature and at ω → Rc ( ω ) = 1 ν F γ γ ln γγ − iω . (9) IV. TUNNELING CONDUCTIVITY
For a calculation of the tunneling conductivity we uselinear response theory in which the tunneling conductiv-ity σ ( V ) at a finite voltage bias V can be presented inthe form of Kubo formula σ ( V ) = e h πeV Im[ χ R ( eV )] , (10)where the retarded response function χ R ( ω ) can be ob-tained by the analytical continuation i Ω n → ω + iδ of χ M ( i Ω n ) that is given by χ M ( i Ω n ) = − β Z β − β dτ e i Ω n τ h T M T ( τ ) T + (0) i . (11) -1 =0.04 K =0.08 K =0.12 K =0.16 K mm- ) m0 T (K) FIG. 4. (Color online) The height of the tunnel conductivitypeak σ m0 for noninteracting electrons and holes on tempera-ture T for different values of Cooper pair decay γ . Here T M is the time-ordering symbol for a imaginary time τ and Ω n = 2 πnT is a bosonic Matsubara frequency.In the system of noninteracting electrons and holes the χ M ( i Ω n ) corresponds to the first diagram on the Fig. 1(b) leading to χ R ( ω ) = | t | Π Rc ( ω ). Hence the tunnelingconductivity for noninteracting electrons and holes σ isgiven by σ ( V ) = e h π | t | eV Im[Π Rc ( eV )] . (12)Its value σ m0 ( γ, T ) at zero bias is given by σ m0 ( γ, T ) = 2 πe h ν F | t | πT Ψ ′ (cid:18)
12 + γ πT (cid:19) , (13)and at low temperatures T ≪ γ it transforms to σ m0 ( γ,
0) = 2 πe h ν F | t | γ . (14)Introduction of the electron-hole Coulomb interactionin the ladder approximation leads to three additionalterms for the tunneling conductivity. The first one corre-sponds to second diagram on Fig.2 (b). It is singular ina vicinity of critical temperature and cannot be reducedto the tunneling conductivity of noninteracting quasipar-ticles with a renormalized spectrum due to Cooper pairfluctuations. The other two terms correspond to renor-malization of single-particle Green functions of electronsand holes. They are not singular in a vicinity of criticaltemperature and can be neglected. The tunneling con-ductivity for interacting electrons and holes σ c is givenby σ c ( V ) = e h π | t | eV Im (cid:20) Π Rc ( eV )1 − U ′ Π Rc ( eV ) (cid:21) . (15) (K)(mV) V HW T =0.04 =0.08 =0.12 =0.16 FIG. 5. (Color online) Half-width of the tunnel conductivitypeak V HW for noninteracting electrons and holes as functionof temperature T for different values of Cooper pair decay γ . The denominator of (15) coincides with that in theCooper propagator (4) and tends to zero in a vicinityof critical temperature T d . Hence the Cooper pair fluc-tuations lead to the critical behavior of tunneling con-ductivity in vicinity of the critical temperature and thequantum critical point. Above critical temperature T d the tunneling conductivity at zero bias is given by σ mc ( γ, T ) = σ m0 ( γ, T ) 1( ν F U ′ ) TT d . (16)In vicinity of the critical temperature it diverges as σ mc ( γ, T ) ∼ ( T − T d ) − ν with the critical index ν = 2.At zero temperature tunneling conductivity is given by σ mc ( γ,
0) = σ m0 ( γ,
0) 1( ν F U ′ ) γγ . (17)It diverges in the vicinity of the quantum phase transitionat γ = γ as σ mc ( γ, ∼ ( γ − γ ) − µ with the critical index µ = 2.The formulas (16) and (17) are the main result of thearticle. The calculated contribution of Cooper pair fluc-tuations to the tunneling conductivity cannot be reducedto the one of noninteracting quasiparticles with renormal-ized spectrum due to Cooper pair fluctuations. It can beinterpreted as the direct contribution of Cooper pair fluc-tuations. Hence the predicted effect of the enhancementof tunneling conductivity in a vicinity of the critical tem-perature and the quantum critical point is the collectiveeffect that can be interpreted as fluctuational internalJosephson effect. V. ANALYSIS AND DISCUSSION
Tunneling conductivity at finite voltage bias and infull-range of temperature and Cooper pair decay was cal- mm- ) mc =0.04 K =0.08 K =0.12 K =0.16 K T (K) FIG. 6. (Color online) The height of the tunnel conductivitypeak σ mc for interacting electrons and holes on temperature T for different values of Cooper pair decay γ . culated numerically according to formulas (5), (12) and(15). The following set of the parameters T = 0 . E F = 5 meV, ν F U ′ = 0 . t = 10 µ eV was used. Theset corresponds to Bi Se TI film with width 10 nm. Thephase diagram of the system is presented on the inset ofFig. 3. If Cooper pair decay rate exceeds critical value γ = 0 .
09 K then the electron-hole pairing is suppressedby a disorder.Calculated tunneling conductivity both for noninter-acting electrons and holes and interacting ones for γ =0 . T = 0 . γ, T ). The peakappears due to restrictions connected with energy andmomentum conservation for tunneling electrons. Suchpeak was predicted and observed also in electron-electronbilayers and it is the peculiarity of a tunneling be-tween two two-dimensional systems. Coulomb interac-tion between electrons and holes considerably enhancesthe tunneling conductivity but does not change qualita-tively its dependence on external bias.Height and half-width of the peak for noninteract-ing electrons are presented on Figs. 4 and 5. Thepeak becomes more prominent with decreasing of Cooperpair decay and temperature. The peaks width is deter-mined by max { γ, T } . The peaks height is determined as1 / max { γ, T } and it is decreasing as 1 /T at T ≫ γ . Theheight and half-width of the peak for interacting electronsand holes are presented on Fig. 6 and Fig. 7, respectively.For γ < γ Coulomb interaction leads to the critical be-havior of the tunneling conductivity in a vicinity of thecritical temperature that we interpret as fluctuational in-ternal Josephson effect. In vicinity of the critical temper-ature height of the peak diverges with the critical index ν = 2 that agrees with the analytic results and the widthof the peak linearly tends to zero. At high temperatures (K) T V HW (mV) FIG. 7. (Color online) Half-width of the tunnel conductivitypeak V HW for interacting electrons and holes as function oftemperature T for different values of Cooper pair decay rate γ . peaks height decreases as 1 / ( T log ( T /T d )). The criticalregion in which tunneling conductivity is considerablyenhanced is of order ∆ T ≈ T d . If the Cooper pair decayexceeds the critical value γ > γ Coulomb interactionconsiderably enhances the height and leads to reductionof the width but does not lead to any singularities. Thepeak becomes more prominent with decreasing of decayand temperature as it does in the model of noninter-acting electrons and holes. The peaks height smoothlydepends on temperature but its maximal value at zerotemperature diverges as function of Cooper pair decay γ in vicinity of quantum critical point at γ . The width ofthe critical region is of order ∆ γ ≈ γ .The peaks width and height smoothly depend on theparameters of the system used for the calculation andlisted above. But a satisfaction of the number of as-sumptions is important for observation of fluctuationalinternal Josephson effect .The model we use here is well applicable in the regimeof weak hybridization t ≪ T in which influence of tun-neling on Cooper pairing can be neglected. In ultrathinTI film the regime of strong hybridization t ≫ T can berealized. In that regime tunneling induces the gap t inthe spectrum of electrons and holes which is considerablelarger than the one due to their Cooper pairing. Criti-cal temperature and Cooper instability are considerablysmoothed in that case. So in that regime we do not ex-pect critical behavior of the tunneling conductivity dueto Cooper pair fluctuations. Our calculations for Bi Se shows that regime of strong hybridization can be realizedin films which width is less then 10 nm.The mean field theory we use here for the descrip-tion of fluctuational internal Josephson effect does notaccount large scale fluctuation of phase of Cooper paircondensate. In two-dimensional superfluids phase fluctu-ations destroy long-range coherence and the transition topaired state at critical temperature T d calculated withinmean field theory is smoothed. Moreover the transi-tion to superfluid state is Berezinskii-Kosterlitz-Thoulesstransition that corresponds to dissociation of vortex-antivortex pairs and which temperature is below T d .Hence the large scale phase fluctuations of Cooper paircondensate can smooth the critical behavior of tunnelconductivity we predict here. But if the size of the sys-tem is comparable with coherence length of Cooper pairfluctuations l c ≈ ~ v F /T the phase fluctuations are unim-portant and mean field theory is well applicable. For T = 0 . l c ∼
10 mkm and we conclude that thedeveloped microscopical theory is applicable for samplesof the corresponding size.The model we use here implies conservation of the mo-mentum of tunneling electron. If the momentum is notconserved the tunneling process creates electron-hole pairwith nonzero total momentum of order l − . Here l T ischaracter length at which tunneling matrix matrix ele-ment t can be considered as constant. Cooper pair isformed by electron and hole with opposite momenta andthe Cooper instability is smoothed if l c ≫ l T . For tun-neling between opposite surfaces of topological insulatorthin film of high crystalline quality momentum conserva-tion can be achieved to remarkable degree.We have shown that electron-hole Coulomb interac-tion considerably enhances the tunneling conductivityin electron-hole bilayer even when the Cooper pairingis suppressed by disorder. The opposite situation takesplace in electron-electron bilayer that also can be realizedin semiconductor quantum well structure, in graphenedouble layer system and in a film of topological insula-tor. Coulomb interaction gives contribution to decay ofelectrons that was analyzed in and to additional seriesof diagrams for the tunneling conductivity. We treatedthe additional diagrams in the ladder approximation (SeeFig.2-b). If they are omitted the tunneling conductivityat zero temperature and at a finite bias V is given by σ ee0 = gt A πe h ν F | t | γ γ γ + ( eV ) , (18)where g is the degeneracy factor of electrons and t A is additional factor that depends on internal nature ofelectrons . The dependence of tunnel conductivity onexternal bias contains prominent peak which becomesmore prominent with decreasing of decay rate γ . If theCoulomb interaction is treated in ladder approximationthe tunneling conductivity is given by σ eec = gt A πe h ν F | t | γ γ γ + (1 + ( ν F U ′ ) )( eV ) . (19)For electron-electron bilayer Coulomb interaction doesnot influences the height of the peak and leads to de-creasing of the width which is insignificant even in thecase of strong interaction ν F U ′ ∼
1. The roles of the interlayer Coulomb interaction in electron-electron bi-layer and electron-hole bilayer are drastically different be-cause the correction to the tunneling conductivity of theprimer is caused by the scattering diagrams in particle-antiparticle channel and the correction to the one ofthe latter is caused by the diagrams in particle-particleCooper channel that contains instability.We have investigated the manifestations of Cooperelectron-hole pairing fluctuations in thin film of topolog-ical insulator on tunneling between its opposite surfaces.The internal fluctuational Josephson effect is general phe-nomenon but each realization of electron-hole bilayer hasits own peculiarities.Dirac points in graphene are situated in corners of firstBrillouin zone. Electron-hole pairing was predicted inthe system of two independently gated graphene layersseparated by dielectric film. In that case orientations ofthe graphene lattices are uncorrelated. The distance be-tween Dirac points of different layers in momentum spaceis of order a − , where a is lattice constant of graphene.The tunneling of electrons between Dirac points is pos-sible if l T ∼ a that corresponds to tunneling throughimpurity states or other defects. The condition l c ≫ l T is well satisfied and the critical behavior of tunneling con-ductivity in double layer graphene system is considerablysmoothed. But if the mutual orientation of graphene lay-ers can be controlled in experiment the presented heretheory is well applicable for that system. The formulas(16),(17) are reasonable and can be easily generalized toadditional spin and valley degree of freedom of electronsand holes. So the fluctuational internal Josephson effectcan also be experimentally investigates in that system.Recently anomalies in drag effect in semiconductordouble well structure that contains spatially separatedelectrons and holes were observed . The analysis ofresults shows that the observed anomalies can be causedby electron-hole pairing not in BCS regime but ratherin regime of BCS-BEC crossover . In that regimeelectron-hole pairing fluctuations also should increasetunneling conductivity of the system but quantitativetheory of the effect is interesting and challenging prob-lem. The developed here microscopical theory of fluc-tuation internal Josephson effect is applicable for thatsystem if the pairing is realized in BCS regime. More-over the formulas (16),(17) are reasonable in that case.In the semiconductor double well structures the con-centrations of electrons and holes can be independentlycontrolled and separate contacts to the layers have beenmade. So in that system tunneling conductivity betweenelectron and hole layers can be measured and the predic-tions of our work can be also addressed.Internal Josephson effect has been observed in quan-tum Hall bilayer at total occupation factor ν T = 1. Butabove critical temperature the dependence of tunnelingconductivity on external bias does not contain any peakdue to non Fermi-liquid behavior of composite electronsand holes . Thus fluctuational effects in that systemare more complicated ones and need separate investiga-tion. It should be noted that critical behavior of tun-neling conductivity has not been observed yet in experi-ments in that system.We have considered influence of electron-hole Cooperpair fluctuations that are precursor of their Cooper pair-ing in topological insulator film on tunneling between itsopposite surface. Cooper pair fluctuations lead to criticalbehavior of tunneling conductivity in vicinity of criticaltemperature with critical index ν = 2. If pairing is sup-pressed by disorder the behavior of tunneling conductiv-ity in vicinity of quantum critical point at zero tempera- ture is also critical with critical index µ = 2. The effectcan be interpreted as fluctuational Josephson effect. ACKNOWLEDGMENTS
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